MATHEMATICAL MODELING OF POPULATION GROWTH OF UGANDA

Transcription

1 KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY Avenue de l armée, P.O. BOX 3900 Kigali Rwanda FACULTY OF SCIENCE DEPARTMENT OF APPLIED MATHEMATICS (STATISTICS OPTION) FINAL YEAR PROJECT MATHEMATICAL MODELING OF POPULATION GROWTH OF UGANDA Project ID: Done by: Epiphanie KAGOYIRE (GS ) and Pacifique ICYINGENEYE (GS ) Supervised by: Prof. Augustus NZOMO Wali TOWARDS A BRIGHTER FUTURE Academic Year 2011

2 DECLARATION We, Epiphanie KAGOYIRE (GS ) and Pacifique ICYINGENEYE (GS ) declare that this project work MATHEMATICAL MODELING OF POPULATION GROWTH OF UGANDA is our original work and has never been presented before for any academic award either in this or other institutions of higher learning for academic publication or any other purpose. Signature:. Signature: Epiphanie KAGOYIRE Pacifique ICYINGENEYE Date:./../ Date:./../ i

3 KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY Avenue de l armée, P.O. BOX 3900 Kigali-Rwanda FACULTY OF SCIENCE DEPARTMENT OF APPLIED MATHEMATICS (STATISTICS OPTION) CERTIFICATE This is to certify that the project work entitled MATHEMATICAL MODELING OF POPULATION GROWTH OF UGANDA is a record of original work done by Epiphanie KAGOYIRE (GS ) and Pacifique ICYINGENEYE (GS ), as partial fulfillment of the requirements for award of the Bachelor of Science Degree in APPLIED MATHEMATICS (STATISTICS OPTION) at KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY during the academic year Signature:.. Signature:.. Prof. Augustus NZOMO Wali SUPERVISOR Prof. Augustus NZOMO Wali HEAD OF DEPARTMENT Date: / /. Date:.././ ii

4 DEDICATION From Epiphanie KAGOYIRE This project work is dedicated to: My husband Jean de Dieu NKURUNZIZA, My mother Marianne MUKARWEGO, My son Maxime NZIZA NYAGATARE. From Pacifique ICYINGENEYE This project work is dedicated to: My mother Liberathe NDUWAMALIYA, My sister Ignaciene UWAYEZU, Jeanne D Arc UWIZEYIMANA and Jean Népomuscène NIYIBIZI. iii

5 ACKNOWLEDGEMENT We are indebted to GOD for his mercy, who with His mighty power protects us during the whole period of our studies. Upon completion of this work, we would like to thank all those who, directly or indirectly, contributed to its outcome. Special thanks go to Rwanda government through the Student Financing Agency for Rwanda (SFAR) which taken us in charge to accomplish our studies. We are also grateful to Kigali Institute of Science and Technology, all Lecturers in faculty of Science, but especially those of Department of Applied Mathematics who fully contributed to our university trainings. We express our deep gratitude to Prof. Augustus NZOMO Wali, who, despite his many duties agreed to lead our work. His guidance, his remarks and relevant suggestions have been to carry out this work. Our thanks go to our colleagues for their advice and encouragement. Finally, to all families and friends, there are no profound words to express our gratitude for the love and support that have given us. It is therefore for lack of a better word that we only say thank you. iv

6 ABSTRACT Uganda is a landlocked country in East Africa. It is bordered on the east by Kenya, north by south Sudan, west by the Democratic Republic of Congo, southwest by Rwanda and to the south by Tanzania. It has an area of 236,040 km 2. Its population is predominated in rural with highest density in the southern regions. Resources of any country play an important role in its population growth. In case of unlimited resources the population grows exponentially without bound but in the environment of limited resources the population exhibit a pattern of growth called logistic growth. The purpose of this project focuses on the application of logistic equation to model the population growth of Uganda using data from 1980 to The data used were collected from International Data Base (IDB) online and were analyzed using MATLAB software. We also used least square method to compute the year when the population will be a half of the value of carrying capacity, population growth rate and the carrying capacity. Population growth of any country depends on the vital coefficients. In the case of Uganda we found that the vital coefficients and are and respectively. Thus the population growth rate of Uganda, according to this model, is 3.6% per annum. This approximated population growth rate compares well with statistically predicted values in literature. We also found that the population of Uganda, 58 years from the year 2010, is expected to be 147,633,806 while the predicted carrying capacity for the population is 295,267,612. To derive more accurate models of population growth, the population can be subdivided into males and females. Other models can be developed by subdividing the population into different age groups. v

7 TABLE OF CONTENTS DECLARATION... i CERTIFICATE... ii DEDICATION... iii ACKNOWLEDGEMENT... iv ABSTRACT...v TABLE OF CONTENTS... vi DEFINITION OF TERMS... vii ABBREVIATIONS...x TABLES... xi FIGURES... xii CHAPTER 1: INTRODUCTION Problem statement Objectives General objective Specific objectives Significance of the study...2 CHAPTER 2: LITERATURE REVIEW...3 CHAPTER 3: METHODOLOGY Study site and data source Data collection method Statistical data analysis method...5 CHAPTER 4: DEVELOPMENT OF THE MODEL...6 CHAPTER 5: RESULTS AND ANALYSIS Interpretation of figure Interpretation of figure CHAPTER 6: CONCLUSION AND RECOMMENDATIONS Conclusion Recommendations...14 REFERENCES...15 APPENDICES...16 vi

8 DEFINITION OF TERMS Population is a group of organisms of the same species inhabiting a given area. Population growth refers to change in the size of a population (which can be either positive or negative) over time, depending on the balance of births and deaths, and it can be quantified as the change in the number of individuals in a population using per unit time for measurement. Population growth rate is the average annual percent change in population, resulting from a surplus (or deficit) of births over deaths and the balance of migrants entering and leaving a country. The rate may be positive or negative. It is a factor in determining how great a burden would be imposed on a country by the changing needs of its people for infrastructure. Population density refers to the number of people in a defined jurisdiction, in relation to the size of the area that they occupy. Obviously, the population density is higher in urban areas than in rural communities. The carrying capacity of a biological species in an environment is the population size of the species that the environment can sustain indefinitely, given the food, habitat, water and other necessities available in the environment. A mathematical model describes a system by a set of variables and a set of equations that establish relationships between the variables. The process of developing a mathematical model is termed mathematical modeling. A differential equation is an equation involving independent and dependent variables and the derivatives or differentials of one or more dependent variables with respect to one or more independent variables. An example of a differential equation is (1) The order of highest order derivative involved in a differential equation is called the order of a differential equation. If a differential equation is involved in only first, second or third derivatives of the dependent variable, it is called a first, second or third order differential equation respectively. Examples: 1 st order differential equation: (2) vii

9 2 nd order differential equation: (3) 3 rd order differential equation: (4) A differential equation in which the dependent variable and all its derivatives present occur in the first degree only and no product of dependent variable and/or derivatives occur is known as a linear differential equation. A differential equation which is not linear is called non linear differential equation. The general form of first order linear differential equation is (5) Where and are function of t or can be constants. If is identically equal to zero, the differential equation is said to be homogeneous. Thus, (6) is an homogeneous equation. A differential equation that is not homogeneous is a non-homogeneous differential equation. A solution of a differential equation is a relation between the dependent and independent variables, not involving the derivatives such that this relation and the derivatives obtained from it satisfy the given differential equation. For example,, where is any constant (7) is a solution of a differential equation (8) because and satisfy the given differential equation. A solution which contains a number of arbitrary constants equal to the order of the differential equation is called the general solution or complete solution of the differential equation. viii

10 A solution obtained from general solution by giving particular values to the constants is called a particular solution. Usually these particular values are given as initial conditions. A solution which cannot be derived from the general solution of the given differential equation is called a singular solution. For example, let equation (8) be a differential equation with initial conditions when. Multiplying both sides of equation (8) by gives, (9) The term can be written as and equation (9) becomes (10) Integrating equation (10) gives, ( is a constant) (11) So that, (12) This equation is a general solution of a differential equation in equation (8). Applying the initial conditions given to equation (12) we get. Substituting this value of c into equation (12), we get the required solution of equation (8) satisfying the given initial conditions (13) This equation is a particular solution of the differential equation in equation (8). ix

11 ABBREVIATIONS KIST: Kigali Institute of Science and Technology IDB: International Data Base km 2 : Square Kilometer x

12 TABLES Table 1: Actual population values...9 Table 2: Actual and predicted population values...11 xi

13 FIGURES Figure 1: Graph of actual and predicted population values against time...12 Figure 2: Graph of predicted population values against time...13 xii

14 CHAPTER 1: INTRODUCTION A mathematical model describes a system by a set of variables and a set of equations that establish relationships between the variables. The process of developing a mathematical model is termed mathematical modeling. It is a process of mimicking reality by using mathematical languages. The examination of population growth is done through observation, experimentation or through mathematical modeling. In this project we will model the population growth of Uganda using Verhulst model. Mathematical models can take many forms, including but not limited to dynamical systems, statistical models and differential equations. These and other types of models can overlap, with a given model involving a variety of abstract structures. First order differential equation governs the growth of various species. A first glance it would seem impossible to model the growth of a species by a differential equation since the population of any species always changes by integer amounts. Hence the population of any species can never be a differentiable function of time. However if a given population is very large and it is suddenly increased by one, then the change is very small compared to the given population, [5]. Thus we make approximation that large populations change continuously and even differentially with time Problem statement Uganda is a country in East Africa. Its population is predominated in rural with highest density in the southern regions. Uganda has substantial natural resources, including fertile soils, regular rainfall, small deposits of copper, gold, and other minerals and recently discovered oil. Agriculture is the most important sector of the economy, employing over 80% of the work force. It has industries processing of agricultural products (cotton ginning, coffee curing), cement production, light consumer goods, textiles. As the population of Uganda is continuously growing, the economy of the country has not been able to keep up with demand for public services. Uganda s population growth rate is worrying, it remains a major challenge to government s efforts to reduce poverty and provide adequate social services like health, education, water and sanitation, housing, food among others. 1

15 In this project we wish to model the population growth of Uganda and the model results will help in predicting future populations. This can be useful for the concerned governmental institutions to plan well for future resource allocations and infrastructure Objectives General objective To create a mathematical model of population growth of Uganda Specific objectives To compute the Vital coefficients, Carrying capacity, Population growth rate Significance of the study This study will help some students to understand applications of differential equations. They will be able to develop and use mathematical models in real life. It shows how researchers can use mathematical models as well as differential equations to solve, or try to solve, real life problems. It will help many people to know the importance of controlling population growth. The model results will help in predicting future populations. This can be useful for the concerned governmental institutions to plan well for future resource allocations and infrastructure. 2

16 CHAPTER 2: LITERATURE REVIEW In 1798, an Englishman Thomas R. Malthus, [3, 4], proposed a mathematical model of population growth which is unconstrained growth, i.e. model in which the population increases in size without bound. It is an exponential growth model governed by a differential equation: (14) Where: represents the time period, represents the population size at time and is the Malthusian factor, is the multiple that determines the growth rate. This model is a simplistic linear equation and is known as Malthusian law of population growth. takes on only integral values and is a discontinuous function of. However may be approximated by a continuous and differentiable function as soon as the number of individuals is large enough. The solution of equation (14) is ( is a constant) (15) If the population of the given species is at time, then satisfy the initial value conditions,. The solution of this initial value conditions is (16) Hence any species satisfying the Malthusian law of population growth grows exponentially with time. The initial-value problem, occurs in many physical theories involving either growth or decay. For example, in physics it provides a model for approximating the remaining amount of a substance which is disintegrating through radioactivity. As noted by Turchin, [6], equation (14) is referred to as The exponential law and is a good candidate for the first principle of population dynamics. His formulation of this principle is as follows: a population will grow (or decline) exponentially as long as the environment experienced by all individuals in the population remains constant. In 1840, a Belgian Mathematician Verhulst, [7], thought that population growth depends not only on the population size but also on how far this size is from its upper limit. 3

17 He modified equation (14) to make the population size proportional to both the previous population and a new term (17) Where and are called the vital coefficients of the population. This term reflects how far the population size is from its upper limit. However, as the population value grows and gets closer to, this new term will become very small and get to zero, providing the right feedback to limit the population growth. Thus the second term models the competition for available resources, which tends to limit the population growth. The modified equation is (18) It is a nonlinear differential equation unlike equation (14) in the sense that one cannot simply multiply the previous population by a factor. In this case the population on the right of equation (18) is being multiplied by itself. This equation is known as Logistic law of population growth. 4

18 CHAPTER 3: METHODOLOGY 3.1. Study site and data source The site of the study in this project was Uganda and the data were collected from IDB Data collection method To achieve the objectives of this project, yearly data of population of Uganda from 1980 to 2010 were collected from IDB online Statistical data analysis method MATLAB software was used to compute the predicted population values and to plot down the graph of these values. This software was also used to plot the graph of actual values. Least square method was used to determine one of the vital coefficients,, carrying capacity and the year when the population will be a half of the value of carrying capacity. 5

19 CHAPTER 4: DEVELOPMENT OF THE MODEL Equation (18) simplifies to (19) Separating the variables, we obtain Taking integrals, we have i.e. Or (20) Using and, we see that Equation (20) becomes Solving for gives (21) If we take the limit of equation (21) as, we get (since ) Next we determine the values of, and by using the least square method. Differentiating equation (21) twice with respect to, gives (22) (23) At the point of inflection this second derivative of when must be equal to zero. This will be so (24) 6

20 Putting we get Solving for gives (25) This is the time when the point of inflection occurs. Let the time when the point of inflexion occurs be. Then becomes. Using this new value of and replacing by, equation (21) becomes (26) Let the coordinates of the actual population values be (, ) and the coordinates of the predicted population values with the same abscissa on the fitted curve be (, P). The error in this case is (P- ). Because some of the actual population data points lie below the predicted values curve while others lie above it, we square (P- ) to ensure that the error is positive. Thus, the total squared error, e, in fitting the curve is given by (27) Equation (27) contains three parameters, and Where. To eliminate, we let (28) (29) Using the value of (27), we have in equation (28) and algebraic properties of inner product to equation 7

21 Where. Therefore, Taking partial derivative of with respect to and equating it to zero, we obtain. This gives (30) (31) Substituting this value of into equation (30), we get (32) This equation is converted into an error function, MATLAB program, [appendix 2(a)], that contains just two parameters, and. Their values are found, MATLAB program, [appendix 2(b)] and used in equation (31) to find the value of, MATLAB program, [appendix 2(c)]. 8

22 CHAPTER 5: RESULTS AND ANALYSIS Table 1: Actual population values Year Population Year Population ,414, ,248, ,725, ,861, ,078, ,502, ,470, ,227, ,919, ,955, ,391, ,690, ,910, ,469, ,520, ,321, ,176, ,233, ,832, ,199, ,455, ,206, ,082, ,262, ,729, ,367, ,424, ,369, ,127, ,398, ,689,516 Using actual population values, their corresponding years from table 1 and MATLAB programs, [appendix 2(a) and 2(b)], we find that the values of and are and respectively. Thus, the value of the population growth rate is approximately 3.6% per annum while the population will be a half of the limiting value in the year Using values of and and MATLAB program [appendix 2(c)], equation (31) gives (33) This is the predicted carrying capacity or limiting value of the population of Uganda. Using equation (22) and value of, we find that (34) 9

23 This value is the other vital coefficient of the population. If we let to correspond to the year 1980, then the initial population will be 12,414,719. Substituting the values of, and into equation (21), we obtain (35) This equation was used to compute the predicted values of population. Using values of, and, equation (25) gives the time at the point of inflection to be (36) Using this value of and equation (35), we get (37) The following table contains the predicted population values and their corresponding actual population values. 10

24 Table 2: Actual and predicted population values Year Actual Population Predicted Population Year Actual Population Predicted Population ,414,719 12,414, ,248,718 21,266, ,725,252 12,845, ,861,011 21,980, ,078,930 13,290, ,502,140 22,716, ,470,393 13,749, ,227,669 23,474, ,919,514 14,224, ,955,822 24,255, ,391,743 14,714, ,690,002 25,061, ,910,724 15,220, ,469,579 25,890, ,520,093 15,743, ,321,962 26,744, ,176,418 16,283, ,233,661 27,623, ,832,384 16,840, ,199,390 28,528, ,455,758 17,414, ,206,503 29,460, ,082,137 18,007, ,262,610 30,418, ,729,453 18,619, ,367,972 31,404, ,424,376 19,251, ,369,558 32,418, ,127,590 19,902, ,398,682 33,460, ,689,516 20,573,841 Below is the graph of actual and predicted population values 11

25 Figure 1: Graph of actual and predicted population values against time Below is the graph of predicted population values. The values were computed using equation (35). 12

26 Figure 2: Graph of predicted population values against time 5.1. Interpretation of figure 1 In figure 1 we see that the actual data points and predicted values are very close to one another. This indicates that error between them is very small Interpretation of figure 2 The curve fitted perfectly well into the Verhulst logistic curve, thus the model is good. The time period before the population reaches half of its limiting value is a period of accelerated growth. After this point, the rate of growth starts to decrease. The value of is the horizontal asymptote of the curve, thus this value is the limiting value of the population. 13

27 CHAPTER 6: CONCLUSION AND RECOMMENDATIONS 6.1. Conclusion The population of Uganda is increasing in time. This increment is due to the reason that many families consider the children as a source of wealth. For that, some children are taken out of schools after only few years; especially girls. They make early marriages and have larger families. In this project, the model accurately fitted the logistic curve. We found that the predicted carrying capacity of the population of Uganda is 295,267,612. Population growth of any country depends on the vital coefficients. In the case of Uganda we found that the vital coefficients and are and respectively. Thus, the population growth rate of Uganda according to this model is approximately 3.6% per annum. According to this model, the population of Uganda will reach a half of its limiting value within 58 years from the year 2010, i.e in the year Recommendations The following are some recommendations: (a). Uganda s high population growth rate remain a major challenge to government s effort to improve and provide adequate social services like education, water and sanitation, housing, food and others; then government should step up civic education on family planning method to reduce population growth rate. (b). The government should make policy of education for all, to avoid early marriages. (c). The government should work towards industrialization of the country, so that some people will get jobs and they will be able to fulfill all needs of their families. (d). Since the reproduction rate in a population usually depends on the number of females more than on the number of males; to derive more accurate models of population growth, the population can be subdivided into males and females. Other models can be developed by subdividing the population into different age groups. (e). The government should work to increase technological development which will affect the increase of its country s ability to support its population. (f). Technological developments, pollution and social trends must be re-evaluated every few years as they have significant influence on the vital coefficients and. 14

28 REFERENCES [1]. Augustus Wali, Doriane Ntubabare, Vedaste Mboniragira, 2011, Mathematical Modeling of Rwanda s Population Growth: Journal of Applied Mathematical Sciences, Vol. 5, no. 53, [2]. International Data Base (IDB): (accessed on April 16 th, 2011). [3]. Malthus, (1798): An Essay on the Principle of population (1 st Edition, plus Excerpts nd US edited by Apple man. Norton critical Editions. [ISBN X] edition), Introduction by Philip Apple man, and associated commentary on Malthus. [4]. Malthus, (1798): An Essay on the Principle of population (1 st edition) with A Summary View (1830), and Introduction by Professor Antony Flew. Penguin Classics ISBN X. [5]. Martin Braun, (1993): Differential Equations and Their Applications (4 th Edition), Springer-Verlag New York, Inc. [6]. Turchin, P. 2001: Does population ecology have general laws? Oikos 94: [7]. Verhulst, P. F., (1838): Notice sur la loi que la population poursuit dans son accroissement. Correspondence Mathématique et Physique. 10: [8]. Zafar Ahsan, (2004): Differential Equations and Their Applications (2 nd Edition), Prentice-Hall of India Private Limited, NewDelhi

29 APPENDICES 1. Uganda map Source: worldmap.org/maps 16

30 2. MATLAB programs (a) MATLAB program used to find error function syms r tk; t=[1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995, 1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010] ; p=[ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ] ; H=1./(1+exp(-r*(t-tk))) t=t' p=p' H=H' e=(p'*p)-((h'*p)^2/(h'*h)) Note that r and tk are replacing and respectively. t and p are time and actual population respectively. (b) MATLAB program used to find the values of r and tk which minimize error function banana=@(x)e; format long [x]=fminsearch(banana,[0.1,2100]) Note that r and tk, in error function, must be replaced by x(1) and x(2) respectively. 0.1 and 2100 are starting points. e is the error function. 17

31 (c) MATLAB program used to find the value of K r=0.0356; tk=2068; t=[1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995, 1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010]; p=[ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ]; H=1./(1+exp(-r*(t-tk))) H=H' p=p' K=(H'*p)/(H'*H) t and p are time and actual population respectively. (d) MATLAB program used to find the predicted values t=0 :30 ; format long P= /(1+( )*(0.965).^t ) t is the time. 18

32 (e) MATLAB program used to plot the graph of actual and predicted values t=0:30; p=[ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ]; format long P= /(1+( )*(0.965).^t ) plot(t,p,'o',t,p) xlabel('time') ylabel('population') t, p and P are time, actual population and predicted population respectively. (f) MATLAB program used to plot the graph of predicted values with extended time t=0:260; format long P= /(1+( )*(0.965).^t ) plot(t,p) xlabel('time') ylabel('population') t and P are time and predicted population respectively. 19